
\section{Problem Definition}
\label{sec:problemdef}

The problem we tackle in this project is to automatically map corresponding
variables (sensors) between source and target domain of a transfer learning problem.
This can happen at the level of the sensors (direct) or at the level of meta-features (indirect) which
we now define.

\begin{mydef}[Sensor]
The sensors in this case are binary: active or not active. In order to
recognize activities they are analyzed for temporal patterns. For instance, a
sequence of sensor readings for ``fridge'' and ``cupboard plates'' and an early
time of the day might indicate that the inhabitant is making breakfast.
\end{mydef}

\begin{mydef}[Meta-feature]
A meta-feature is a collection of individual sensors. In the context of this work ``meta-feature'' is synonymous to ``group of sensors'' 
(we use these terms interchangeably). It generally evaluates to True if
at least one of its members (sensors) evaluates to True. Note that meta-features can be
constructed in different ways which we will discuss in section 5.
\end{mydef}

In our project we specifically focus on the use case of behavior recognition in
a household environment. The features of the learning problem correspond to a
number of binary sensors installed in the environment.

The testing data consists of sensor firings which includes the id of
the sensor, the time when the sensor starts to fire (becomes True), and the time
when it stops to fire (becomes False). Time is represented by calendar time,
including the date and time of day. In order to make better use of sparse
measurements we abstract them as follows.

\begin{mydef}[Abstract Measurement Representation]
A sensor measurement is represented by time-of-day of the starting time and the
duration of the activation.
\end{mydef}

Transfer learning can be successfully applied only if a mapping between the
sensors of two domains can be established. For this objective we need to
establish a generic and abstract representation by which we can describe or
model a sensor. Concretely, we model each sensor by creating a sensor
profile and a relational profile.

\begin{mydef}[Sensor Profile]
The sensor profile of a sensor is a distribution over the space of abstract
measurements. It characterizes a sensor in terms of when it most likely fires
or activates, and how long each activation is likely to last.
\end{mydef}

\begin{mydef}[Relational Profile]
The relational profile of a sensor relates the sensor to other sensors within a domain
to show if there is synchronized behavior.
\end{mydef}

Essentially, the sensor profile models the sensor itself and the relational
profile gives a context in terms of other sensors.

We now define three different problems involved in mapping sensors across domains.

\begin{myprob}[Direct Sensor Mapping]
\label{prob:directmapping}
Given a source domain $S$ and a target domain $T$ find a direct association between the
sensors of both domains. Find a mapping:  $M : S \rightarrow T$. In general an injective function is preferred.
\end{myprob}

The main point of direct sensor mapping is that with the given training mapping
$Z$ we have identities of sensors across domains and can thus reliably merge the
training data. This enables us to create more general sensor models that are
less prone to noise and which also decrease differences between the
distributions of the two training domains, corresponding to different behavior
of people. The transfer of knowledge can then be carried out along mappings
defined by such generic models.

Another approach of matching sensors between domains is to go one level higher
and to try matching meta-features. This involves the construction of an abstract layer.
Mapping is performed at the level of this abstract layer. In order to allow a mapping at this
abstract layer, meta-features should be constructed. The automatic construction of meta-features
is a domain specific problem. 

\begin{myprob}[Meta-feature Construction]
\label{prob:mfconstruction}
Given a domain $D$, group sensors within domain $D$ according to a similarity measure.
The meta-feature is an abstract representation of a group of similar sensors (depending on the similarity measure).
Meta-features should be carefully constructed in order not to impede the process of activity recognition (that is performed after knowledge transfer) .
\end{myprob}

\begin{myprob}[Indirect Sensor Mapping Using Meta-features]
\label{prob:mfmapping}
Given a source domain $S$ and a target domain $T$ find a direct association between the
meta-features of both domains. Find a mapping:  $M : S \rightarrow T$. In general an injective function is preferred.
\end{myprob}

Throughout all of the experiments we will assume the following:

\begin{description}
 \item[Assumption]  Inhabitants of different households have similar living patterns.
\end{description}

Without this assumption, transfer of knowledge in a subsequent stadium would not be favorable. 
Therefore finding a variable mapping would serve no goal. An interesting related question that comes to mind is; 
if we cannot find a variable mapping automatically, would transfer of knowledge serve it's purpose?
If the variables across domains cannot be related automatically, this is an indication that the living patterns of inhabitants between houses differ.
If we transfer activity related knowledge between houses of people with different living patterns there is the risk of negative transfer.


